AIME 试题汇编(2000-2015).pdf

2000 AIME Problems AIME Problems 2000 I March 28th 1Find the least positive integer n such that no matter how 10nis expressed as the product of any two positive integers at least one of these two integers contains the digit 0 2Let u and v be integers satisfying 0 v u Let A u v let B be the refl ection of A across the line y x let C be the refl ection of B across the y axis let D be the refl ection of C across the x axis and let E be the refl ection of D across the y axis The area of pentagon ABCDE is 451 Find u v 3In the expansion of ax b 2000 where a and b are relatively prime positive integers the coeffi cients of x2and x3are equal Find a b 4The diagram shows a rectangle that has been dissected into nine non overlapping squares Given that the width and the height of the rectangle are relatively prime positive integers fi nd the perimeter of the rectangle 5Each of two boxes contains both black and white marbles and the total number of marbles in the two boxes is 25 One marble is taken out of each box randomly The probability that both marbles are black is 27 50 and the probability that both marbles are white is m n where m and n are relatively prime positive integers What is m n Contributors 4everwise white horse king88 joml88 rrusczyk 2000 AIME Problems 6For how many ordered pairs x y of integers is it true that 0 x y 0 Find m n r 14Every positive integer k has a unique factorial base expansion f1 f2 f3 fm meaning that k 1 f1 2 f2 3 f3 m fm where each fiis an integer 0 fi i and 0 fm Given that f1 f2 f3 fj is the factorial base expansion of 16 32 48 64 1968 1984 2000 fi nd the value of f1 f2 f3 f4 1 j 1fj 15Find the least positive integer n such that 1 sin45 sin46 1 sin47 sin48 1 sin133 sin134 1 sinn These problems are copyright c Mathematical Association of America http maa org Contributors 4everwise white horse king88 joml88 rrusczyk 2001 AIME Problems AIME Problems 2001 I March 27th 1Find the sum of all positive two digit integers that are divisible by each of their digits 2 A fi nite set S of distinct real numbers has the following properties the mean of S 1 is 13 less than the mean of S and the mean of S 2001 is 27 more than the mean of S Find the mean of S 3Find the sum of the roots real and non real of the equation x2001 1 2 x 2001 0 given that there are no multiple roots 4In triangle ABC angles A and B measure 60 degrees and 45 degrees respec tively The bisector of angle A intersects BC at T and AT 24 The area of triangle ABC can be written in the form a b c where a b and c are positive integers and c is not divisible by the square of any prime Find a b c 5An equilateral triangle is inscribed in the ellipse whose equation is x2 4y2 4 One vertex of the triangle is 0 1 one altitude is contained in the y axis and the length of each side is pm n where m and n are relatively prime positive integers Find m n 6 A fair die is rolled four times The probability that each of the fi nal three rolls is at least as large as the roll preceding it may be expressed in the form m n where m and n are relatively prime positive integers Find m n 7Triangle ABC has AB 21 AC 22 and BC 20 Points D and E are located on AB and AC respectively such that DE is parallel to BC and contains the center of the inscribed circle of triangle ABC Then DE m n where m and n are relatively prime positive integers Find m n 8Call a positive integer N a 7 10 double if the digits of the base 7 representation of N form a base 10 number that is twice N For example 51 is a 7 10 double because its base 7 representation is 102 What is the largest 7 10 double Contributors joml88 4everwise rrusczyk 2001 AIME Problems 9In triangle ABC AB 13 BC 15 and CA 17 Point D is on AB E is on BC and F is on CA Let AD p AB BE q BC and CF r CA where p q and r are positive and satisfy p q r 2 3 and p2 q2 r2 2 5 The ratio of the area of triangle DEF to the area of triangle ABC can be written in the form m n where m and n are relatively prime positive integers Find m n 10Let S be the set of points whose coordinates x y and z are integers that satisfy 0 x 2 0 y 3 and 0 z 4 Two distinct points are randomly chosen from S The probability that the midpoint of the segment they determine also belongs to S is m n where m and n are relatively prime positive integers Find m n 11In a rectangular array of points with 5 rows and N columns the points are numbered consecutively from left to right beginning with the top row Thus the top row is numbered 1 through N the second row is numbered N 1 through 2N and so forth Five points P1 P2 P3 P4 and P5 are selected so that each Piis in row i Let xibe the number associated with Pi Now renumber the array consecutively from top to bottom beginning with the fi rst column Let yibe the number associated with Piafter the renumbering It is found that x1 y2 x2 y1 x3 y4 x4 y5 and x5 y3 Find the smallest possible value of N 12A sphere is inscribed in the tetrahedron whose vertices are A 6 0 0 B 0 4 0 C 0 0 2 and D 0 0 0 The radius of the sphere is m n where m and n are relatively prime positive integers Find m n 13In a certain circle the chord of a d degree arc is 22 centimeters long and the chord of a 2d degree arc is 20 centimeters longer than the chord of a 3d degree arc where d 120 The length of the chord of a 3d degree arc is m n centimeters where m and n are positive integers Find m n 14A mail carrier delivers mail to the nineteen houses on the east side of Elm Street The carrier notices that no two adjacent houses ever get mail on the same day but that there are never more than two houses in a row that get no mail on the same day How many diff erent patterns of mail delivery are possible 15The numbers 1 2 3 4 5 6 7 and 8 are randomly written on the faces of a regular octahedron so that each face contains a diff erent number The probability that no two consecutive numbers where 8 and 1 are considered to Contributors joml88 4everwise rrusczyk 2001 AIME Problems be consecutive are written on faces that share an edge is m n where m and n are relatively prime positive integers Find m n II April 10th 1Let N be the largest positive integer with the following property reading from left to right each pair of consecutive digits of N forms a perfect square What are the leftmost three digits of N 2Each of the 2001 students at a high school studies either Spanish or French and some study both The number who study Spanish is between 80 percent and 85 percent of the school population and the number who study French is between 30 percent and 40 percent Let m be the smallest number of students who could study both languages and let M be the largest number of students who could study both languages Find M m 3Given that x1 211 x2 375 x3 420 x4 523 and xn xn 1 xn 2 xn 3 xn 4when n 5 fi nd the value of x531 x753 x975 4Let R 8 6 The lines whose equations are 8y 15x and 10y 3x contain points P and Q respectively such that R is the midpoint of PQ The length of PQ equals m n where m and n are relatively prime positive integers Find m n 5A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive Consider sets 4 5 6 n of consecutive positive integers all of whose ten element subsets have the triangle property What is the largest possible value of n Contributors joml88 4everwise rrusczyk 2001 AIME Problems 6Square ABCD is inscribed in a circle Square EFGH has vertices E and F on CD and vertices G and H on the circle The ratio of the area of square EFGH to the area of square ABCD can be expressed as m n where m and n are relatively prime positive integers and m n Find 10n m 7Let PQR be a right triangle with PQ 90 PR 120 and QR 150 Let C1be the inscribed circle Construct ST with S on PR and T on QR such that ST is perpendicular to PR and tangent to C1 Construct UV with U on PQ and V on QR such that UV is perpendicular to PQ and tangent to C1 Let C2be the inscribed circle of RST and C3the inscribed circle of QUV The distance between the centers of C2and C3can be written as 10n What is n 8A certain function f has the properties that f 3x 3f x for all positive real values of x and that f x 1 x 2 for 1 x 3 Find the smallest x for which f x f 2001 9Each unit square of a 3 by 3 unit square grid is to be colored either blue or red For each square either color is equally likely to be used The probability of obtaining a grid that does not have a 2 by 2 red square is m n where m and n are relatively prime positive integers Find m n 10How many positive integer multiples of 1001 can be expressed in the form 10j 10i where i and j are integers and 0 i j 99 11Club Truncator is in a soccer league with six other teams each of which it plays once In any of its 6 matches the probabilities that Club Truncator will win lose or tie are each 1 3 The probability that Club Truncator will fi nish the season with more wins than losses is m n where m and n are relatively prime positive integers Find m n 12Given a triangle its midpoint triangle is obtained by joining the midpoints of its sides A sequence of polyhedra Pi is defi ned recursively as follows P0is a regular tetrahedron whose volume is 1 To obtain Pi 1 replace the midpoint triangle of every face of Piby an outward pointing regular tetrahedron that has the midpoint triangle as a face The volume of P3is m n where m and n are relatively prime positive integers Find m n 13In quadrilateral ABCD BAD ADC and ABD BCD AB 8 BD 10 and BC 6 The length CD may be written in the form m n where m and n are relatively prime positive integers Find m n Contributors joml88 4everwise rrusczyk 2001 AIME Problems 14There are 2n complex numbers that satisfy both z28 z8 1 0 and z 1 These numbers have the form zm cos m isin m where 0 1 2 2n 360 and angles are measured in degrees Find the value of 2 4 2n 15Let EFGH EFDC and EHBC be three adjacent square faces of a cube for which EC 8 and let A be the eighth vertex of the cube Let I J and K be the points on EF EH and EC respectively so that EI EJ EK 2 A solid S is obtained by drilling a tunnel through the cube The sides of the tunnel are planes parallel to AE and containing the edges IJ JK and KI The surface area of S including the walls of the tunnel is m n p where m n and p are positive integers and p is not divisible by the square of any prime Find m n p These problems are copyright c Mathematical Association of America http maa org Contributors joml88 4everwise rrusczyk 2002 AIME Problems AIME Problems 2002 I March 26th 1Many states use a sequence of three letters followed by a sequence of three digits as their standard license plate pattern Given that each three letter three digit arrangement is equally likely the probability that such a license plate will contain at least one palindrome a three letter arrangement or a three digit arrangement that reads the same left to right as it does right to left is m n where m and n are relatively prime positive integers Find m n 2The diagram shows twenty congruent circles arranged in three rows and en closed in a rectangle The circles are tangent to one another and to the sides of the rectangle as shown in the diagram The ratio of the longer dimension of the rectangle to the shorter dimension can be written as 1 2 p q where p and q are positive integers Find p q 3Jane is 25 years old Dick is older than Jane In n years where n is a positive integer Dick s age and Jane s age will both be two digit number and will have the property that Jane s age is obtained by interchanging the digits of Dick s age Let d be Dick s present age How many ordered pairs of positive integers d n are possible 4 Consider the sequence defi ned by ak 1 k2 k for k 1 Given that am am 1 an 1 1 29 for positive integers m and n with m y x y r xr rxr 1y r r 1 2 xr 2y2 r r 1 r 2 3 xr 3y3 What are the fi rst three digits to the right of the decimal point in the decimal representation of 102002 1 10 7 8Find the smallest integer k for which the conditions 1 a1 a2 a3 is a non decreasing sequence of positive integers 2 an an 1 an 2for all n 2 3 a9 k are satisfi ed by more than one sequence 9Harold Tanya and Ulysses paint a very long picket fence Harold starts with the fi rst picket and paints every hth picket Tanya starts with the second picket and paints everth tth picket and Ulysses starts with the third picket and paints every uth picket Call the positive integer 100h 10t u paintable when the triple h t u of positive integers results in every picket being painted exaclty once Find the sum of all the paintable integers 10In the diagram below angle ABC is a right angle Point D is on BC and AD bisects angle CAB Points E and F are on AB and AC respectively so that AE 3 and AF 10 Given that EB 9 and FC 27 fi nd the integer closest to the area of quadrilateral DCFG Contributors joml88 chess64 4everwise paladin8 JesusFreak197 Elemennop tetrahedr0n frt t0rajir0u rrusczyk 2002 AIME Problems A B C D E F G 10 27 3 9 11Let ABCD and BCFG be two faces of a cube with AB 12 A beam of light emanates from vertex A and refl ects off face BCFG at point P which is 7 units from BG and 5 units from BC The beam continues to be refl ected off the faces of the cube The length of the light path from the time it leaves point A until it next reaches a vertex of the cube is given by m n where m and n are integers and n is not divisible by the square of any prime Find m n 12Let F z z i z i for all complex numbers z 6 i and let zn F zn 1 for all positive integers n Given that z0 1 137 i and z2002 a bi where a and b are real numbers fi nd a b 13In triangle ABC the medians AD and CE have lengths 18 and 27 respectively and AB 24 Extend CE to intersect the circumcircle of ABC at F The area of triangle AFB is m n where m and n are positive integers and n is not divisible by the square of any prime Find m n 14A set S of distinct positive integers has the following property for every integer x in S the arithmetic mean of the set of values obtained by deleting x from S is an integer Given that 1 belongs to S and that 2002 is the largest element of S what is the greatet number of elements that S can have 15Polyhedron ABCDEFG has six faces Face ABCD is a square with AB 12 face ABFG is a trapezoid with AB parallel to GF BF AG 8 and GF 6 and face CDE has CE DE 14 The other three faces are ADEG BCEF and EFG The distance from E to face ABCD is 12 Given that EG2 p q r where p q and r are positive integers and r is not divisible by the square of any prime fi nd p q r II Contributors joml88 chess64 4everwise paladin8 JesusFreak197 Elemennop tetrahedr0n frt t0rajir0u rrusczyk 2002 AIME Problems April 9th 1Given that 1 x and y are both integers between 100 and 999 inclusive 2 y is the number formed by reversing the digits of x and 3 z x y How many distinct values of z are possible 2Three vertices of a cube are P 7 12 10 Q 8 8 1 and R 11 3 9 What is the surface area of the cube 3It is given that log6a log6b log6c 6 where a b and c are positive integers that form an increasing geometric sequence and b a is the square of an integer Find a b c 4Patio blocks that are hexagons 1 unit on a side are used to outline a garden by placing the blocks edge to edge with n on each side The diagram indicates the path of blocks around the garden when n 5 Contributors joml88 chess64 4everwise paladin8 JesusFreak197 Elemennop tetrahedr0n frt t0rajir0u rrusczyk 2002 AIME Problems If n 202 then the area of the garden enclosed by the path not including the path itself is m 3 2 square units where m is a positive integer Find the remainder when m is divided by 1000 5Find the sum of all positive integers a 2n3m where n and m are non negative integers for which a6is not a divisor of 6a 6Find the integer that is closest to 1000 P10000 n 3 1 n2 4 7It is known that for all positive integers k 12 22 32 k2 k k 1 2k 1 6 Find the smallest positive integer k such that 12 22 32 k2is a multiple of 200 8Find the least positive integer k for which the equation 2002 n k has no integer solutions for n The notation x means the greatest integer less than or equal to x 9Let S be the set 1 2 3 10 Let n be the number of sets of two non empty disjoint subsets of S Disjoint sets are defi ned as sets that have no common elements Find the remainder obtained when n is divided by 1000 10 While fi nding the sine of a certain angle an absent minded professor failed to notice that his calculator was not in the correct angular mode He was lucky to get the right answer The two least positive real values of x for which the sine of x degrees is the same as the sine of x radians are m n and p q where m n p and q are positive integers Find m n p q 11 Two distinct real infi nite geometric series each have a sum of 1 and have the same second term The third term of one of the series is 1 8 and the second term of both series can be written in the form m n p where m n and p are positive integers and m is not divisible by the square of any prime Find 100m 10n p 12A basketball player has a constant probability of 4 of making any given shot independent of previous shots Let anbe the ratio of shots made to shots attempted after n shots The probability that a10 4 and an 4 for all n Contributors joml88 chess64 4everwise paladin8 JesusFreak197 Elemennop tetrahedr0n frt t0rajir0u rrusczyk 2002 AIME Problems such that 1 n 9 is given to be paqbr sc where p q r and s are primes and a b and c are positive integers Find p q r s a b c 13In triangle ABC point D is on BC with CD 2 and DB 5 point E is on AC with CE 1 and EA 32 AB 8 and AD and BE intersect at P Points Q and R lie on AB so that PQ is parallel to CA and PR is parallel to CB It is given that the ratio of the area of triangle PQR to the area of triangle ABC is m n where m and n are relatively prime positive integers Find m n 14The perimeter of triangle APM is 152 and the angle PAM is a right angle A circle of radius 19 with center O on AP is drawn so that it is tangent to AM and PM Given that OP m n where m and n are relative